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Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

✍ Scribed by F. Otto; C. Villani

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 244 KB
Volume: 173
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1999.3557

No coin nor oath required. For personal study only.

✦ Synopsis

We show that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal. 6, 587 600) for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension and optimal in certain cases. The proofs are based on partial differential equations and an interpolation inequality involving the Wasserstein distance, the entropy functional, and the Fisher information.

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